## Why is e^(ln x)=x? (O Level Math/ A Level Math Tuition)

Why is $\boxed{e^{\ln x}=x}$?

This formula will be useful for some questions in O Level Additional Maths, or A Level H2 Maths.

There are two ways to show or prove this, first we can let

$y=e^{\ln x}$

Taking natural logarithm (ln) on both sides, we get

$\ln y=\ln x\ln e=\ln x$

So $y=x$. Substitute the very first equation and we get $e^{\ln x}=x$. 🙂

Alternatively, we can view $e^x$ and $\ln x$ as inverse functions of each other. So, we can let $f(x)=e^x$ and $f^{-1}(x)=\ln x$. Then, $e^{\ln x}=f(f^{-1})(x)=x$ by definition of inverse functions. This may be a better way to remember the result. 🙂

The above method of inverse functions can be used to remember $\ln (e^x)=x$ too.